Theorem clwwlknun 16365

Description: The set of closed walks of fixed length N in a simple graph G is the union of the closed walks of the fixed length N on each of the vertices of graph G. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 3-Mar-2022.) (Proof shortened by AV, 28-Mar-2022.)

Hypothesis

Ref	Expression
clwwlknun.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
clwwlknun	|- ( G e. USGraph -> ( N ClWWalksN G ) = U_ x e. V ( x (ClWWalksNOn ` G ) N ) )

Distinct variable groups:   x, G   x, N   x, V

Proof of Theorem clwwlknun

Dummy variables y i are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 3979	. . 3 |- ( y e. U_ x e. V ( x (ClWWalksNOn ` G ) N ) <-> E. x e. V y e. ( x (ClWWalksNOn ` G ) N ) )
2		isclwwlknon 16354	. . . . 5 |- ( y e. ( x (ClWWalksNOn ` G ) N ) <-> ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) )
3	2	rexbii 2540	. . . 4 |- ( E. x e. V y e. ( x (ClWWalksNOn ` G ) N ) <-> E. x e. V ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) )
4		simpl 109	. . . . . 6 |- ( ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) -> y e. ( N ClWWalksN G ) )
5	4	rexlimivw 2647	. . . . 5 |- ( E. x e. V ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) -> y e. ( N ClWWalksN G ) )
6		clwwlknun.v	. . . . . . . . 9 |- V = (Vtx ` G )
7		eqid 2231	. . . . . . . . 9 |- (Edg ` G ) = (Edg ` G )
8	6, 7	clwwlknp 16341	. . . . . . . 8 |- ( y e. ( N ClWWalksN G ) -> ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) )
9	8	anim2i 342	. . . . . . 7 |- ( ( G e. USGraph /\ y e. ( N ClWWalksN G ) ) -> ( G e. USGraph /\ ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) ) )
10	7, 6	usgrpredgv 16122	. . . . . . . . . . . . 13 |- ( ( G e. USGraph /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) -> ( (lastS ` y ) e. V /\ ( y ` 0 ) e. V ) )
11	10	ex 115	. . . . . . . . . . . 12 |- ( G e. USGraph -> ( { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) -> ( (lastS ` y ) e. V /\ ( y ` 0 ) e. V ) ) )
12		simpr 110	. . . . . . . . . . . 12 |- ( ( (lastS ` y ) e. V /\ ( y ` 0 ) e. V ) -> ( y ` 0 ) e. V )
13	11, 12	syl6com 35	. . . . . . . . . . 11 |- ( { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) -> ( G e. USGraph -> ( y ` 0 ) e. V ) )
14	13	3ad2ant3 1047	. . . . . . . . . 10 |- ( ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) -> ( G e. USGraph -> ( y ` 0 ) e. V ) )
15	14	impcom 125	. . . . . . . . 9 |- ( ( G e. USGraph /\ ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) ) -> ( y ` 0 ) e. V )
16		simpr 110	. . . . . . . . . . . 12 |- ( ( ( G e. USGraph /\ ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) ) /\ x = ( y ` 0 ) ) -> x = ( y ` 0 ) )
17	16	eqcomd 2237	. . . . . . . . . . 11 |- ( ( ( G e. USGraph /\ ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) ) /\ x = ( y ` 0 ) ) -> ( y ` 0 ) = x )
18	17	biantrud 304	. . . . . . . . . 10 |- ( ( ( G e. USGraph /\ ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) ) /\ x = ( y ` 0 ) ) -> ( y e. ( N ClWWalksN G ) <-> ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) ) )
19	18	bicomd 141	. . . . . . . . 9 |- ( ( ( G e. USGraph /\ ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) ) /\ x = ( y ` 0 ) ) -> ( ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) <-> y e. ( N ClWWalksN G ) ) )
20	15, 19	rspcedv 2915	. . . . . . . 8 |- ( ( G e. USGraph /\ ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) ) -> ( y e. ( N ClWWalksN G ) -> E. x e. V ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) ) )
21	20	adantld 278	. . . . . . 7 |- ( ( G e. USGraph /\ ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) ) -> ( ( G e. USGraph /\ y e. ( N ClWWalksN G ) ) -> E. x e. V ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) ) )
22	9, 21	mpcom 36	. . . . . 6 |- ( ( G e. USGraph /\ y e. ( N ClWWalksN G ) ) -> E. x e. V ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) )
23	22	ex 115	. . . . 5 |- ( G e. USGraph -> ( y e. ( N ClWWalksN G ) -> E. x e. V ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) ) )
24	5, 23	impbid2 143	. . . 4 |- ( G e. USGraph -> ( E. x e. V ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) <-> y e. ( N ClWWalksN G ) ) )
25	3, 24	bitrid 192	. . 3 |- ( G e. USGraph -> ( E. x e. V y e. ( x (ClWWalksNOn ` G ) N ) <-> y e. ( N ClWWalksN G ) ) )
26	1, 25	bitr2id 193	. 2 |- ( G e. USGraph -> ( y e. ( N ClWWalksN G ) <-> y e. U_ x e. V ( x (ClWWalksNOn ` G ) N ) ) )
27	26	eqrdv 2229	1 |- ( G e. USGraph -> ( N ClWWalksN G ) = U_ x e. V ( x (ClWWalksNOn ` G ) N ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2202   A.wral 2511   E.wrex 2512   {cpr 3674   U_ciun 3975   ` cfv 5333  (class class class)co 6028   0cc0 8075   1c1 8076   + caddc 8078   - cmin 8392  ..^cfzo 10422  ♯chash 11083  Word cword 11162  lastSclsw 11207  Vtxcvtx 15936  Edgcedg 15981  USGraphcusgr 16078   ClWWalksN cclwwlkn 16327  ClWWalksNOncclwwlknon 16350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-2o 6626  df-er 6745  df-map 6862  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-7 9249  df-8 9250  df-9 9251  df-n0 9445  df-z 9524  df-dec 9656  df-uz 9800  df-fz 10289  df-fzo 10423  df-ihash 11084  df-word 11163  df-ndx 13148  df-slot 13149  df-base 13151  df-edgf 15929  df-vtx 15938  df-iedg 15939  df-edg 15982  df-umgren 16018  df-usgren 16080  df-clwwlk 16316  df-clwwlkn 16328  df-clwwlknon 16351

This theorem is referenced by: (None)